Regularization of currents and entropy

Dinh, Tien-Cuong; Sibony, Nessim

doi:10.1016/j.ansens.2004.09.002

Cited by 107 publications

(175 citation statements)

References 18 publications

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“…0 C and g 0 and L 1 loc in X such that k Q u l k Ä g for all l (see the theorem of Fischer-Riesz in [12], Proposition 13.11.4 (ii)). The regularization obtained is therefore of the same type of the Dinh-Sibony regularization (see [13]). …”

Section: Closed Regularization With Negative Part Converging To 0 Formentioning

confidence: 99%

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Regularization of closed positive currents and intersection theory

Méo

2017

Complex Manifolds

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Abstract:We prove the existence of a closed regularization of the integration current associated to an effective analytic cycle, with a bounded negative part. By means of the King formula, we are reduced to regularize a closed differential form with L 1 loc coefficients, which by extension has a test value on any positive current with the same support as the cycle. As a consequence, the restriction of a closed positive current to a closed analytic submanifold is well defined as a closed positive current. Lastly, given a closed smooth differential .q 0 ; q 0 /-form on a closed analytic submanifold, we prove the existence of a closed .q 0 ; q 0 /-current having a restriction equal to that differential form. After blowing up we deal with the case of a hypersurface and then the extension current is obtained as a solution of a linear differential equation of order 1.

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Section: Closed Regularization With Negative Part Converging To 0 Formentioning

confidence: 99%

“…Recall also the regularization theorem of Dinh-Sibony (see [13]) which applies to any q but only claims the existence of a closed regularization with a negative part that is bounded in the L 1 loc sense. This theorem asserts precisely the existence of a constant C dependent of !…”

Section: Introductionmentioning

confidence: 99%

Regularization of closed positive currents and intersection theory

Méo

2017

Complex Manifolds

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“…p X g for some constant A > 0 independent of T . The following semi-regularization of currents was proved by Sibony and the first author in [6], [7]. We need the following lemma.…”

Section: Positive Closed Currentsmentioning

confidence: 99%

“…It was shown in [6], [7] that OE p .f n / 1=n converges to a constant d p .f / which is the dynamical degree of order p of f . Note that the main difficulty here is that in general we do not have .f nCs / D .f n / B .f s / on cohomology classes.…”

Section: Dynamical Degreesmentioning

confidence: 99%

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Comparison of dynamical degrees for semi-conjugate meromorphic maps

Dinh¹,

Nguyên²

2011

Comment. Math. Helv.

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Decay of correlations and the central limit theorem for meromorphic maps

Dinh

Sibony

2006

Comm Pure Appl Math

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Let f be a dominant meromorphic self-map of large topological degree on a compact Kähler manifold. We give a new construction of the equilibrium measure µ of f and prove that µ is exponentially mixing. As a consequence, we get the central limit theorem in particular for Hölder-continuous observables, but also for noncontinuous observables. DECAY OF CORRELATIONS 755THEOREM 1.1 The measure µ is exponentially mixing in the following sense: for every > 0 and 0 < α ≤ 1, there exists a constant A ,α > 0 such thatfor all n ≥ 0, for every function ψ ∈ L ∞ (µ), and for every Hölder-continuous function ϕ of order α. If a real-valued Hölder-continuous function ϕ is not a coboundary and verifies µ, ϕ = 0, then it satisfies the CLT theorem.We recall in Section 2 the Gordin-Liverani theorem and in Section 3 some properties of the Sobolev space W 1,2 . The space of test functions W 1,2 * and a subspace W 1,2 * * are introduced in Section 4. This is the key point of the method. The new space W 1,2 * seems to be useful for complex dynamics in several variables. In complex dimension 1 it coincides with the space W 1,2 . In higher dimensions it takes into account the fact that not all currents of bidegree (1, 1) are closed. It enjoys the composition properties under meromorphic maps, useful for a space of observables. In Section 5, we give the new construction of µ, and in Section 6 its statistical properties. Note that the geometric decay of correlations for Hénon maps was recently proved in [6] using the dd c -method. Like the dd c -method, the d-method can be applied to some meromorphic correspondences and to random iteration [7].Notation. We will use different subspaces of L 1 (X ). Most of them carry a canonical (quasi) norm. For the space C 0 (X ) of continuous functions we use the supnorm, for the space Lip(X ) of Lipschitz functions the norm · L 1 + · Lip where ϕ Lip := sup x =y |ϕ(x) − ϕ(y)| dist(x, y) −1 , and for the Hölder space C α (X ) the norm ϕ C α := ϕ L 1 + sup x =y |ϕ(x) − ϕ(y)| dist(x, y) −α . We use the (quasi)norm · E + · F for the intersection E ∩ F of E and F and write L p instead of L p (X ) when there is no confusion. The L p norm of a form is the sum of L p norms of its coefficients for a fixed atlas of X . The topology that we consider is a weak topology. That is, ϕ n → ϕ in E if ϕ n → ϕ in the sense of distributions and ( ϕ n E ) is bounded. The continuity of linear operators : E → F is with respect to these topologies of E and F. The inclusion map E ⊂ L 1 (X ) is always bounded for the associated (quasi)-norms and is continuous in our sense.

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Regularization of currents and entropy

Cited by 107 publications

References 18 publications

Regularization of closed positive currents and intersection theory

Regularization of closed positive currents and intersection theory

Comparison of dynamical degrees for semi-conjugate meromorphic maps

Decay of correlations and the central limit theorem for meromorphic maps

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